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Traveling wave solutions of a highly nonlinear shallow water equation

  • * Corresponding author: Anna Geyer

    * Corresponding author: Anna Geyer 
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  • Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.

    Mathematics Subject Classification: Primary: 35Q35, 58F17; Secondary: 34C37.


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  • Figure 1.  A selection of some traveling wave solutions of (1). The waves on the left side from top to bottom are of the following types: smooth periodic, peaked periodic, cusped periodic, periodic with peaked crests and cusped troughs, periodic with peaked crests and troughs, composite, composite with plateaus. Right side top to bottom: smooth solitary, peaked solitary, cusped solitary, wavefront, compactly supported anticusp, multi-crest with decay, multi-peak with decay.

    Figure 2.  (a) the graphs of the functions $\alpha_2$ [bold], $\alpha_1$ [plain] and $\bar u$ [dashed]; (b) the graphs of $K_{\alpha_2}$ [bold], $K_{\alpha_1}$ [plain] and $K_0$ [dashed], cf. (24) and (29).

    Figure 3.  The graph of $\varphi$ for (a) $c>\bar c$ and (b) $c = \bar c$.

    Figure 4.  The graph of $\varphi$ for increasing values of $c \in (-\infty, \bar c)$.

    Figure 5.  In (5a) we sketch a phase portrait representing scenario Ⅲ in Table 1, which yields a smooth solitary and smooth periodic traveling waves as illustrated in (5b).

    Figure 6.  Sketches of phase portraits for scenario Ⅰ, i.e. $K <K_0(c)$.

    Figure 7.  Sketches of phase portraits for scenario Ⅱ, i.e. $K = K_0(c)$

    Figure 8.  Sketches of the phase portraits of scenario Ⅲ: (8a)-(8d) yield smooth periodic waves, (8a) and (8d) yield smooth solitary waves, (8b) yields peaked solitary waves, (8c) yields peaked periodic and-provided that $c\in (c_0,c_1)$-both periodic and solitary cusped traveling waves.

    Figure 9.  Sketches of peaked (9a) and cusped (9b) traveling waves.

    Figure 10.  Sketches of phase portraits in scenario Ⅳ, i.e. $K = K_{\alpha_1}(c)$.

    Figure 11.  Some examples of composite waves constructed from orbits corresponding to scenario Ⅳ: smooth (11a) and peaked (11b) traveling wave solutions with plateaus at height $\alpha_1$, smooth (11c) and peaked (11d) multi-crested solutions with decay, and smooth (11e) and non-smooth (11f) compactons.

    Figure 12.  Sketches of phase portraits in scenario Ⅴ, i.e. $K\in(K_{\alpha_1}(c),K_{\alpha_2}(c))$.

    Figure 13.  Traveling waves constructed from orbits corresponding to the phase portrait in Scenario Ⅴ, Fig. 12a: Fig. 13a shows solitary and periodic anti-cusped waves. Fig. 13b shows composite waves: a steep wave front and a periodic composition.

    Figure 14.  Traveling waves constructed from orbits corresponding to the phase portrait in Scenario Ⅴ, Fig. 12b: Fig. 14a shows solitary and periodic anti-peaked waves. Fig. 14b shows composite waves: a wave front and a periodic composition.

    Figure 15.  Sketches of phase portraits in scenario Ⅵ, i.e. $K = K_{\alpha_2}(c)$.

    Figure 16.  Sketches of phase portraits in scenario Ⅶ, i.e. $K>K_{\alpha_2}(c)$.

    Figure 17.  In Fig. 17a we see sketches of an anti-cusped and an anti-peaked solitary wave taking the constant value $\alpha_2$ outside a bounded interval-they correspond to the dark blue lines in Fig. 15a and 15b, respectively. The first image in Fig. 17b shows a periodic wave with peaked crests and cusped troughs. In the second sketch in Fig. 17b we see a periodic wave with peaked crests and troughs. These waves correspond to the dark blue lines in Fig. 16a and 16b, respectively. In Fig. 17c we see an example of a composite wave constructed from orbits of different energy levels in Fig. 16b.

    Figure 18.  Sketches of phase portraits in scenarios Ⅰ and Ⅱ.

    Figure 19.  Sketches of phase portraits in scenarios Ⅲ and Ⅳ.

    Figure 20.  Sketches of the phase portrait corresponding to scenario Ⅴ, i.e. $K>K_{\alpha}$.

    Figure 21.  Sketches of Cantor waves with peak elements (a) and cusp elements (b).

    Table 1.  A list of all possible scenarios for the ordering of fixed points on the horizontal axis. Here s stands for saddle, c for center and n means that the Jacobi matrix at the fixed point is nilpotent.

    scenario parameter order relation fixed points and type
    $K < K_0(c)$ - -
    $K = K_0(c)$ $\bar u <\alpha_1 <\alpha_2 $ $ (\bar u,0)$ n
    $K_0(c) <K <K_{\alpha_1}(c)$ $u_1 < u_2<\alpha_1 <\alpha_2$ $(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$
    $K = K_{\alpha_1}(c)$ $u_1 < u_2=\alpha_1 <\alpha_2$ $(u_1,0)$ $\mathrm s$, $(\alpha_1,0)$ $\mathrm n$
    $K_{\alpha_1}(c) < K < K_{\alpha_2}(c)$ $u_1 <\alpha_1 < u_2 <\alpha_2$ $(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm s$
    $K = K_{\alpha_2}(c)$ $u_1 <\alpha_1 < u_2 = \alpha_2$ $(u_1,0)$ $\mathrm s$, $(\alpha_2,0)$ $\mathrm n$
    $K > K_{\alpha_2}(c)$ $u_1 <\alpha_1 <\alpha_2 <u_2$ $(u_1,0)$ $\mathrm s$, $(u_2,0)$ $\mathrm c$
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